Question

below, ( m angle 3 = 43 ^ { circ } ). find ( m angle 1 ), ( m angle 2 ), and ( m angle 4 ).

Explanation:

Step1: Find \( m\angle1 \)

Vertical angles are equal. \( \angle1 \) and \( \angle3 \) are vertical angles, so \( m\angle1 = m\angle3 \).
Since \( m\angle3 = 43^\circ \), then \( m\angle1 = 43^\circ \).

Step2: Find \( m\angle2 \)

\( \angle2 \) and \( \angle3 \) are supplementary (they form a linear pair), so their measures add up to \( 180^\circ \).
Let \( m\angle2 = x \). Then \( x + 43^\circ = 180^\circ \).
Solving for \( x \), we get \( x = 180^\circ - 43^\circ = 137^\circ \). So \( m\angle2 = 137^\circ \).

Step3: Find \( m\angle4 \)

\( \angle4 \) and \( \angle3 \) are supplementary (they form a linear pair), or \( \angle4 \) and \( \angle2 \) are vertical angles.
Using vertical angles: \( \angle4 \) and \( \angle2 \) are vertical angles, so \( m\angle4 = m\angle2 = 137^\circ \).
Or using supplementary angles with \( \angle3 \): \( m\angle4 + 43^\circ = 180^\circ \), so \( m\angle4 = 180^\circ - 43^\circ = 137^\circ \).

Answer:

\( m\angle1 = \boldsymbol{43}^\circ \)
\( m\angle2 = \boldsymbol{137}^\circ \)
\( m\angle4 = \boldsymbol{137}^\circ \)